Frequencies Of Square Plates Research Articles

Analytical Solution for the Three-Dimensional Vibration of a Rectangular Functionally Graded Material Plate with Two Simply Supported Opposite Faces

Analytical solutions based on three-dimensional (3D) elasticity for the vibrations of functionally graded material (FGM) plates are valuable for assessing the validity and accuracy of various plate theories and numerical approaches. Few benchmark 3D analytical solutions for the vibrations of FGM plates are available in the literature. In this study, analytical solutions based on Fourier series and 3D elasticity were developed for the first time for the vibrations of FGM rectangular plates with two simply supported opposite edge faces. The distributions of the properties of FGMs through the thickness follow a simple power law. The proposed solutions were validated by conducting comprehensive convergence studies on the vibration frequencies of square plates with different thickness-to-side ratios and boundary conditions as well as comparisons with published results. The benchmark nondimensional frequencies were tabulated for plates with free boundary conditions on the top and bottom faces and six combinations of boundary conditions on the other two faces. Moreover, the effects of aspect ratio and gradient index on the vibration frequencies of FGM plates were investigated. The influence of the thickness ratio of the FGM layer to the homogenous layer on the vibration frequencies of sandwich plates with FGM face sheets and a homogeneous core was also studied.

Effect of Shape of Cut-out on Natural Frequency of Square Plate

In this paper, the vibration analysis of a square plate with different cut-out shapes with the same area is investigated. Seven square plates of 150 mm x 150 mm size having central cut-out with all outer edges clamped (C-C-C-C) were considered for the study. Different shapes, namely triangle, square, pentagon, hexagon, heptagon, octagon and circle of the same cross-sectional area of 706.86 mm2 were cut out centrally on each square plate. The modal analyses were carried out using finite element package ANSYS Workbench 15.0. The effect of the shape of cut-out on the natural frequency of the plate was studied. It is observed that the values of natural frequencies are increased by increasing the number of edges of the cut-out.

Assumed strain finite element for natural frequencies of bending plates

PurposeThis paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation.Design/methodology/approachAfter the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem.FindingsThe formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions.Originality/valueThe present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.

Natural frequencies of an inhomogeneous square thin plate

Plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. For free vibration frequencies of a plate, which thickness and/or Young’s modulus depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a square plate with parameters linearly or parabolically depend on the radial coordinate, are examined. The double frequencies of square plates with similar edge support of all edges are of special interest, since any variation of the thickness or stiffness causes some loss of symmetry one may expect the split of double frequencies. The asymptotic formulas permit to determine, which of two equal unperturbed frequencies corresponding to wave numbers n and m increases faster with the small parameter. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.

Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates.

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

Effect of crack location on vibration analysis of partially cracked isotropic and FGM micro-plate with non-uniform thickness: An analytical approach

An analytical model is presented for nonlinear vibration analysis of variable thickness, thin isotropic and functionally graded rectangular micro-plate containing a partial crack located within the centre line of the plate. Linear and parabolic thickness variation is considered in one or both the in-plane directions of the plate. The thickness variation is such that the volume of the plate is equal to that of the uniform thickness plate. The continuous line crack is parallel to one of the edges of the plate. The equations of motion are derived using the equilibrium principle based on Classical Plate Theory while the size effect is incorporated using the modified couple stress theory. The partial crack is represented by bending moment and in-plane force according to the simplified line spring model. The effect of in-plane forces is considered by employing the Berger's formulation. The derived governing equation is converted into a cubic nonlinear Duffing equation by employing Galerkin's method. The effect of nonlinearity is established by deriving the frequency response equation for the cracked variable thickness plate using the method of multiple scales. The nonlinear frequency response curves show the phenomenon of bending hardening or softening. The influence of crack length, crack location, internal material length scale parameter, boundary conditions, gradient index, unidirectional and bidirectional taper constant on the fundamental frequency of square plate is demonstrated. Similar to uniform thickness plate, it is found that the presence of crack affects the vibration characteristics of variable thickness plate. When compared to uniform thickness plate, the effect of crack can be reduced by varying the thickness of the plate. Thus, it is concluded that in a given volume, it is better to employ variable thickness plate as far as the vibration characteristics are considered.

Natural frequencies of square plates with reinforced central holes subjected to inplane loads

The natural frequencies and the elastic buckling loads of square plates, containing reinforced square holes and subjected to inplane loads, are determined using the finite element method. The reinforcing beams which are located at the four edges of the hole are either square or rectangular in cross section. To determine the appropriate out-of-plane stiffness and mass matrices, a rectangular non-conforming bending element is employed. This element has three degrees of freedom at each of its four corner nodes. The finite element method is also used to calculate the inplane stress distribution prior to buckling. This inplane element is rectangular and is based on strain assumptions. The element also includes the inplane rotation as well as the two translational displacements as the three degrees of freedom at each of its four corner nodes. To model the behaviour of the reinforcement, a three-dimensional exact beam element is used. For this element, the degrees of freedom for out-of-plane displacement are consistent with the non-conforming bending element, and the inplane degrees of freedom are consistent with the inplane strain-based element. In the present paper the natural frequencies and the corresponding modes of vibration are investigated when the reinforced plates are subjected to uniform uniaxial, biaxial and shear loads. The variation of the natural frequencies with various sizes of reinforcement is obtained for plates subjected to a range of magnitude of inplane loading.

Natural frequencies of rectangular plates with free edges

This note presents vibration analysis of isotropic rectangular plates with free edges by the Rayleigh-Ritz method with B-spline functions. To show the accuracy of the present method, the results are compared with existing results based on other numerical methods and found to be in good agreement. Accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions. The effects of Poisson's ratio on natural frequencies of square plates with free edges are also investigated.

Vibrations of thin rectangular plates with arbitrarily oriented stiffeners

Free vibration of plates with arbitrarily oriented stiffeners are studied using high precision plate bending and stiffener elements. Good convergence of frequency values for coarse mesh is demonstrated. Natural frequencies of square plates with various arrangement of stiffeners are determined for both simply supported and clamped boundary conditions.

Free vibrations of square plantes with stiffened square openings

A grid frame work model is extended to obtain the natural frequencies of square plates with stiffened square openings. This technique eliminates the use of fictitious points at the re-entrant corner point, junction point, etc. on the stiffened opening. Numerical results are presented for the first time for the cases of plates with simply supported and clamped boundary conditions in order to show the convergence and versatility of the method. Also, this investigation provides a general feeling about the changes in fundamental frequencies that occur when stiffeners are introduced.

Fundamental Frequencies of Simply Supported but Partially Clamped Square Plates

This paper deals with the fundamental vibration of a simply supported but partially clamped square plate. The energy method is applied here, which was proposed in the former paper for solving the general problems of statical deflections, stationary vibrations, bucklings, etc. of rectangular plates. Numerical calculations are caurried out for fourteen cases, two of which are containd in the former paper, and five diagrams are obtained which give the fundamental frequencies of square plates under typical boundary conditions.